Abstract

The Geometrization Theorem of Thurston and Perelman provides a roadmap to understanding topology in dimension 3 via geometric means. Specifically, it states that every closed 3-manifold has a decomposition into geometric pieces, and the zoo of these geometric pieces is quite constrained: each is built from one of eight homogeneous 3-dimensional Riemannian model spaces (called the Thurston geometries).

In this talk, we will approach the question of “what does a 3-manifold look like” from the perspective of geometrization. Through animations of simple examples in dimensions 2 and 3 we review what it means to put a (complete, homogeneous) geometric structure on a manifold, and construct an example admitting each of the Thurston geometries.

Using software written in collaboration with Remi Coulon, Sabetta Matsumoto and Henry Segerman, we will explore these manifolds ``from the inside'' - that is, simulating the view one would have in such a space by raytracing along geodesics. Finally we will touch on how to re-assemble these geometric pieces and understand an “inside view” of general 3-manifolds.

Occurrences

I’ve given a version of this talk to several audiences, often at the math-department-colloquium level. Here’s a short selection:

• Cornell University
• University of Florida
• McGill University
• Gathering for Gardner, Celebration of Mind
• University of Arkansas

Here’s a recording of the talk as given virtually at Cornell, during the Cornell Topology Festival.

And here’s a presentation aimed at a more general audience on similar material, given for Gathering for Gardner, online.

Slides

Here are slides from a recent version of this talk!